Boundedness from below in the real Weinberg three-Higgs-doublet model

Abstract

We study the boundedness-from-below (BFB) conditions of the quartic potential in the real Weinberg three-Higgs-doublet model (3HDM). Unlike an analysis restricted to neutral Higgs fields, a complete stability test must also allow for charge-breaking directions. We show that the problem can be reduced to a finite set of 28 stationary configurations, 14 neutral and 14 charged, each of which can be treated using a linear system of size at most four. The main simplification comes from the one-dimensional kernel of the Gram matrix for charged configurations, while the phase equations reduce to two possibilities: aligned phases or phases forming a closed triangle. We further show how to deal exactly with singular choices of the couplings by introducing an auxiliary deformation and taking an algebraic limit. For rational couplings, the normalized minimum is rational and an attaining field configuration can be constructed algebraically. We discuss analytical examples, exact checks, and comparisons with numerical minimization and StableWein. Our result applies to arbitrary real quartic couplings; it does not cover complex phase coefficients or settle the full-potential stability problem when the quartic is flat.